### EXPLORING THE MEANING OF EXPONENTS

#### S. Kastberg

Consider the process of dividing a line segment into two equal parts by constructing the midpoint. Record the number of midpoints constructed for each iteration, the number of segments, and the length of each segment in a table like the one below, or in a spreadsheet, or in your calculator.

 Number of midpoints number of segments length of each segment 0 1 2 3 4 5

The segment below is one unit long. Use this segment as the basis of your investigation and fill in your table.

What patterns do you observe in your table?

What is the relationship between the entries in the number of segments column? How many segments could be formed if 10 midpoints were constructed? How many segment could be formed if n (where n is any non-negative integer) midpoints were constructed?

What is the relationship between the entries in the length of each segment column? How long would each segment be if 10 midpoints were constructed? How long would each segment be if n (where n is any non-negative integer) midpoints were constructed?

What is the relationship between the entries in the number of midpoints column?

It is hard to compare the entries in the number of segments column since the counts are based on segments of different lengths. Is it possible to restructure this experiement so that the counts are all based on segments of the same length?

Now consider the process of doubling the length of a given segment.

For example: Consider the following segment, AB, of length one unit.

If we constructed a segment AC whose length was twice that of AB then how long would the new segment be?

Easy! 2 units.

Repeat the construction.

Record the number of doubles constructed, the total number of segments, and the length of each segment in a table like the one below, or in a spreadsheet, or in your calculator.

 Number of doubles Number of segments length of each segment 0 1 2 3 4 5

What are the similarities and differences between this table and the one you constructed for the midpoints?

In this table what is the relationship between the entries in the number of segments column? How many segments would their be after 10 doubles had been constructed? How many segments would their be after n doubles had been constructed?

Now suppose that you wanted to extend your table to include finding the midpoint as well as doubling. Keep in mind that we want the length of each segment to stay the same. How would you do this? What would your table look like? What would finding the midpoint mean in terms of number of doubles?

I have included a GSP sketch to help you a bit if you are stuck. Click HERE to see it. Then fill in the table again.

 Number of doubles Number of segments length of each segment 0 1 2 3 4 5

Now can you write each of the number of segments entries using exponents?

 Number of doubles Number of segments in exponential form Number of segments 0 1 2 3 4 5

What mathematical operation would you use to travel up the number of doubles column? What mathematical operation would you use to travel down the number of doubles column?

What mathematical operation would you use to travel up the number of segments in exponential form column? What mathematical operation would you use to travel down the number of segments in exponential form column?

If 10 doubles were constructed, how many segments of one unit each would you have? If you have one thirty-second of a segment how many doubles were constructed?

Reflecting on patterns

Explain what you think the exponents explored in this activity mean?

Explain what the number 2 had to do with this activity?

Using doubling, is it ever possible to get 0 segments? If so how, if not how close can you get?

If you were trying to get 0 segments, how might you do it?